Maximum Matchings via Glauber Dynamics

In this paper we study the classic problem of computing a maximum cardinality matching in general graphs $G = (V, E)$. The best known algorithm for this problem till date runs in $O(m \sqrt{n})$ time due to Micali and Vazirani \cite{MV80}. Even for general bipartite graphs this is the best known running time (the algorithm of Karp and Hopcroft \cite{HK73} also achieves this bound). For regular bipartite graphs one can achieve an $O(m)$ time algorithm which, following a series of papers, has been recently improved to $O(n \log n)$ by Goel, Kapralov and Khanna (STOC 2010) \cite{GKK10}. In this paper we present a randomized algorithm based on the Markov Chain Monte Carlo paradigm which runs in $O(m \log^2 n)$ time, thereby obtaining a significant improvement over \cite{MV80}. We use a Markov chain similar to the \emph{hard-core model} for Glauber Dynamics with \emph{fugacity} parameter $\lambda$, which is used to sample independent sets in a graph from the Gibbs Distribution \cite{V99}, to design a faster algorithm for finding maximum matchings in general graphs. Our result crucially relies on the fact that the mixing time of our Markov Chain is independent of $\lambda$, a significant deviation from the recent series of works \cite{GGSVY11,MWW09, RSVVY10, S10, W06} which achieve computational transition (for estimating the partition function) on a threshold value of $\lambda$. As a result we are able to design a randomized algorithm which runs in $O(m\log^2 n)$ time that provides a major improvement over the running time of the algorithm due to Micali and Vazirani. Using the conductance bound, we also prove that mixing takes $\Omega(\frac{m}{k})$ time where $k$ is the size of the maximum matching.Comment: It has been pointed to us independently by Yuval Peres, Jonah Sherman, Piyush Srivastava and other anonymous reviewers that the coupling used in this paper doesn't have the right marginals because of which the mixing time bound doesn't hold, and also the main result presented in the paper. We thank them for reading the paper with interest and promptly pointing out this mistak

Approximation algorithms for two-state anti-ferromagnetic spin systems on bounded degree graphs

In a seminal paper [10], Weitz gave a deterministic fully polynomial approximation scheme for counting exponentially weighted independent sets (which is the same as approximating the partition function of the hard-core model from statistical physics) in graphs of degree at most d, up to the critical activity for the uniqueness of the Gibbs measure on the in nite d-regular tree. More recently Sly [8] (see also [1]) showed that this is optimal in the sense that if there is an FPRAS for the hard-core partition function on graphs of maximum degree d for activities larger than the critical activity on the in nite d-regular tree then NP = RP. In this paper we extend Weitz's approach to derive a deterministic fully polynomial approximation scheme for the partition function of general two-state anti-ferromagnetic spin systems on graphs of maximum degree d, up to the corresponding critical point on the d-regular tree. The main ingredient of our result is a proof that for two-state anti-ferromagnetic spin systems on the d-regular tree, weak spatial mixing implies strong spatial mixing. This in turn uses a message-decay argument which extends a similar approach proposed recently for the hard-core model by Restrepo et al [7] to the case of general two-state anti-ferromagnetic spin systems

Correlation Decay up to Uniqueness in Spin Systems

We give a complete characterization of the two-state anti-ferromagnetic spin systems which are of strong spatial mixing on general graphs. We show that a two-state anti-ferromagnetic spin system is of strong spatial mixing on all graphs of maximum degree at most \Delta if and only if the system has a unique Gibbs measure on infinite regular trees of degree up to \Delta, where \Delta can be either bounded or unbounded. As a consequence, there exists an FPTAS for the partition function of a two-state anti-ferromagnetic spin system on graphs of maximum degree at most \Delta when the uniqueness condition is satisfied on infinite regular trees of degree up to \Delta. In particular, an FPTAS exists for arbitrary graphs if the uniqueness is satisfied on all infinite regular trees. This covers as special cases all previous algorithmic results for two-state anti-ferromagnetic systems on general-structure graphs. Combining with the FPRAS for two-state ferromagnetic spin systems of Jerrum-Sinclair and Goldberg-Jerrum-Paterson, and the very recent hardness results of Sly-Sun and independently of Galanis-Stefankovic-Vigoda, this gives a complete classification, except at the phase transition boundary, of the approximability of all two-state spin systems, on either degree-bounded families of graphs or family of all graphs.Comment: 27 pages, submitted for publicatio

The Ising Partition Function: Zeros and Deterministic Approximation

We study the problem of approximating the partition function of the ferromagnetic Ising model in graphs and hypergraphs. Our first result is a deterministic approximation scheme (an FPTAS) for the partition function in bounded degree graphs that is valid over the entire range of parameters $\beta$ (the interaction) and $\lambda$ (the external field), except for the case $\vert{\lambda}\vert=1$ (the "zero-field" case). A randomized algorithm (FPRAS) for all graphs, and all $\beta,\lambda$, has long been known. Unlike most other deterministic approximation algorithms for problems in statistical physics and counting, our algorithm does not rely on the "decay of correlations" property. Rather, we exploit and extend machinery developed recently by Barvinok, and Patel and Regts, based on the location of the complex zeros of the partition function, which can be seen as an algorithmic realization of the classical Lee-Yang approach to phase transitions. Our approach extends to the more general setting of the Ising model on hypergraphs of bounded degree and edge size, where no previous algorithms (even randomized) were known for a wide range of parameters. In order to achieve this extension, we establish a tight version of the Lee-Yang theorem for the Ising model on hypergraphs, improving a classical result of Suzuki and Fisher.Comment: clarified presentation of combinatorial arguments, added new results on optimality of univariate Lee-Yang theorem

Spatial mixing and the connective constant: Optimal bounds

We study the problem of deterministic approximate counting of matchings and independent sets in graphs of bounded connective constant. More generally, we consider the problem of evaluating the partition functions of the monomer-dimer model (which is defined as a weighted sum over all matchings where each matching is given a weight γ^(|V| –2|M|) in terms of a fixed parameter γ called the monomer activity) and the hard core model (which is defined as a weighted sum over all independent sets where an independent set I is given a weight γ^(|I|) in terms of a fixed parameter γ called the vertex activity). The connective constant is a natural measure of the average degree of a graph which has been studied extensively in combinatorics and mathematical physics, and can be bounded by a constant even for certain unbounded degree graphs such as those sampled from the sparse Erdös-Rényi model (n, d/n). Our main technical contribution is to prove the best possible rates of decay of correlations in the natural probability distributions induced by both the hard core model and the monomer-dimer model in graphs with a given bound on the connective constant. These results on decay of correlations are obtained using a new framework based on the so-called message approach that has been extensively used recently to prove such results for bounded degree graphs. We then use these optimal decay of correlations results to obtain FPTASs for the two problems on graphs of bounded connective constant. In particular, for the monomer-dimer model, we give a deterministic FPTAS for the partition function on all graphs of bounded connective constant for any given value of the monomer activity. The best previously known deterministic algorithm was due to Bayati, Gamarnik, Katz, Nair and Tetali [STOC 2007], and gave the same runtime guarantees as our results but only for the case of bounded degree graphs. For the hard core model, we give an FPTAS for graphs of connective constant Δ whenever the vertex activity λ λ_c(Δ) would imply that NP=RP [Sly, FOCS 2010]. The previous best known result in this direction was a recent paper by a subset of the current authors [FOCS 2013], where the result was established under the suboptimal condition λ < λc(Δ + 1). Our techniques also allow us to improve upon known bounds for decay of correlations for the hard core model on various regular lattices, including those obtained by Restrepo, Shin, Vigoda and Tetali [FOCS 11] for the special case of ℤ^2 using sophisticated numerically intensive methods tailored to that special case